The concept of minimal dissipation and the identification of work in autonomous systems: A view from classical statistical physics

TL;DR

The paper investigates how to define thermodynamic work for open, possibly strongly coupled systems within an autonomous S+E framework, using projection-operator methods and the concept of minimal dissipation. By constructing a Zwanzig-type projection with a canonical equilibrium weight, the conservative drift of the reduced dynamics is shown to be governed by a Hamiltonian of mean force $H_S^*$, linking the reduced dynamics to the thermodynamic free energy via $F_S=-k_B T\ln Z_S^*$. In quasi-static limits with a large heat-capacity environment, this approach reproduces the conventional work relation $W=\Delta F$, while highlighting that the particular inner product choice crucially determines the outcome and may diverge from minimal-dissipation prescriptions in general. The authors also discuss limitations when extending to non-factorizing quantum environments and emphasize that, for factorizing equilibria, a parallel classical-to-quantum translation can be made, but no universal procedure exists for all systems.

Abstract

Recently, the concept of minimal dissipation has been brought forward as a means to define work performed on open quantum systems [Phys. Rev. A 105, 052216 (2022)]. We discuss this concept from the point of view of projection operator formalisms in classical statistical physics. We analyse an autonomous composite system which consists of a system and an environment in the most general sense (i.e. we neither impose conditions on the coupling between system and environment nor on the properties of the environment). One condition any useful definition of work needs to fulfill is that it reproduces the thermodynamic notion of work in the limit of weak coupling to an environment that has infinite heat capacity. We propose a projection operator route to a definition of work that reaches this limit and we discuss its relation to minimal dissipation.